Journal of Applied Polymer Science, Vol.67, No.1, 119-129, 1998
Elasticity theory and crosslinking of reinforced rubber
For filler fraction C, molecular weight M(c) between crosslinks, and equilibrium swelling v(r), the Flory function M(c) = F(v(r)) was corrected for a hard fraction Ch = (1 + beta)C, beta C being rubber occluded within the primary filler structure. However, a small effect of a Graphon filler was unjustifiably attributed to rubber stretched hard by swelling; later research wrongly attempted to estimate from F(v(r))/F(v(c)) where v(c) considers the hard filler as rubber. By avoiding these mistakes, and with 1/Fo(v(r)) as unfilled crosslinking, Blanchard's constraint equations for linkage reinforcement and reinforced crosslinking 1/M(c) are now simplified to
Alternatively, 1/M(c) can be obtained from the theoretical modulus G = F/(alpah - 1/alpha(2)), by the stress F at extension ratio = 2 following two very different prestretches, alpah(b)≫ 2. The choice of 100% strain (alpha = 2) is to minimize low-strain Mooney-Rivlin deviation from simple rubber theory and to avoid particle contact effects. The choice alpha(b) ≫ alpha ≫ 2 is to avoid stress upturn as alpha -> alpha(b). Then, for two prestretches alpha(b) = 3 and alpha(b) = 4.5 or 5 (400%), corresponding prestresses S(1) and S(2), moduli G(1) and G(2), and force per linkage factors X(1) = alpha(b)S(1)/G(1)(2/3) and X(2) = alpha(b)S(2)/G(2)(2/3), the primary modulus G(*) is
Because k = 0.276 for all rubbers and fillers, the secondary modulus G(r) is known from (G(1) - G(2))/(F(X(1)) - F(X(2))). Hence,
Here (1 + V) allows for network dilution by filler volume V per milliliter of rubber, C = V/(1 + V), and beta C is rubber occluded within the particle aggregate structure of carbon blacks. The measured (effective) crosslinking 1/M(c) is obtained by omitting (1 - beta C)/(1 + beta C) from the above equations. The structure parameter beta might be determined from G(r) using the present test prescription and modern furnace blacks with negligible to high structure.